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The principle of relativity and the De Broglie relation
Julio Güémez∗
Department of Applied Physics, Faculty of Sciences,
University of Cantabria, Av. de Los Castros, 39005 Santander, Spain
Manuel Fiolhais†
Physics Department and CFisUC, University of Coimbra, 3004-516 Coimbra, Portugal
Luis A. Fernández‡
Department of Mathematics, Statistics and Computation,
Faculty of Sciences, University of Cantabria,
Av. de Los Castros, 39005 Santander, Spain
(Dated: February 5, 2016)
Abstract
The De Broglie relation is revisited in connection with an ab initio relativistic description of
particles and waves, which was the same treatment that historically led to that famous relation. In
the same context of the Minkowski four-vectors formalism, we also discuss the phase and the group
velocity of a matter wave, explicitly showing that, under a Lorentz transformation, both transform
as ordinary velocities. We show that such a transformation rule is a necessary condition for the
covariance of the De Broglie relation, and stress the pedagogical value of the Einstein-MinkowskiLorentz relativistic context in the presentation of the De Broglie relation.
1
I.
INTRODUCTION
The motivation for this paper is to emphasize the advantage of discussing the De Broglie
relation in the framework of special relativity, formulated with Minkowski’s four-vectors, as
actually was done by De Broglie himself.1 The De Broglie relation, usually written as
λ=
h
,
p
(1)
which introduces the concept of a wavelength λ, for a “material wave” associated with
a massive particle (such as an electron), with linear momentum p and h being Planck’s
constant. Equation (1) is typically presented in the first or the second semester of a physics
major, in a curricular unit of introduction to modern physics. At that stage, students are
generally not yet acquainted with the relativistic formalism (or even with relativity at all).
Therefore we accept the way in which the De Broglie relation is usually taught, but we
claim that, in a later stage, the relation should be revisited in a full relativistic framework,
such as the one pedagogically presented in this paper. For instance, in a second course in
electromagnetism or in quantum mechanics (wherein special relativity may appear again, this
quantum example can be presented). In this way, students may get a new physical insight
on the interrelations in physics, in particular between relativity and quantum mechanics.
The supposed unfamiliarity with a relativistic formalism is probably the reason why
typical university introductory physics textbooks do not stress that the De Broglie relation connecting wave to particle properties, (expressed either by p~ = ~~k or E = ~ω), are
co-related intrinsically relativistic expressions.2 (In these expressions, p~ is the particle momentum, E is its energy, while ~k and ω are its wavenumber and its angular frequency, and
~ = h/(2π).) In quantum mechanics textbooks, the De Broglie relation is usually presented
in its most simplified form,3 our Eq. (1). On the other hand, basic textbooks on special
relativity typically do not present De Broglie relations (although exceptions can be found4 ).
The “wave-particle duality” concept was actually introduced by Planck when he related
the energy of a stationary electromagnetic wave, of frequency ν, in thermal equilibrium with
the walls of a cavity (blackbody), with the quantum of energy E = hν times an integer
number. Such an idea was reinforced by Einstein when he assigned the linear momentum
pphoton =
2
hν
,
c
(2)
to a photon with frequency ν and energy Ephoton = hν,5 though the fact that light carries linear momentum (besides energy) was already established by the Maxwell’s theory of
electromagnetism.
Now, using the relativistic expressions for the energy and linear momentum of a particle
with inertia m, moving with velocity ~v in a certain inertial frame S, namely E = γmc2 and
p~ = γm~v , where γ = (1 − β 2 )−1/2 and β = v/c, one obtains p~ = E~v /c2 . For a photon,
v = c and, using Planck’s relation E = hν, one arrives at Eq. (2). Several experiments using
light, such as diffraction and interference experiments, showed that light behaves as a wave
of wavelength λ = c/ν. In other experiments, such as those involving Compton scattering,
the photoelectric effect, etc. could only be interpreted using the concept of a photon,6 a
particle with linear momentum p~photon .
Hence, at the beginning of the twentieth century the dual nature of light was unquestionable. Louis De Broglie, in the second decade of last century, was inspired to generalize this
idea, introducing the “wave-particle duality” for massive particles,2 relating the wavelength
of the matter wave and the linear momentum of the particle as expressed by Eq. (1). The
validity of that equation was later on confirmed by Davisson and Germer in their famous
diffraction experiment of electrons on a nickel crystal. Equation (1) is also consistent with
the resonance condition for the Bohr’s atom (quantization of angular momentum).7
The wavelength and the frequency of a plane wave are related through λν = vphase , where
vphase is the so-called phase velocity (i.e. the velocity at which a fixed phase point in a wave
moves through space). The same concept of phase velocity applies to a De Broglie wave.
The energy of a particle of mass m, moving with velocity v, is E = γmc2 . Using Planck’s
equation E = hν, the frequency for this material wave is ν = γmc2 /h. Finally, writing down
the wavelength as λ = h/(γmv), by using the same expression λν = vphase one recognizes
that
c2
.
(3)
v
The way this result was obtained makes it clear that the phase velocity for a De Broglie
vphase =
material wave is an intrinsically relativistic concept. One notes that the De Broglie phase
velocity is larger than c,8 but this is not an issue, because there is no observable associated
with the wave’s phase, which does not carry any physical information.9 Rather, we know
that information cannot be transferred faster than the so-called signal velocity, which is the
speed of the front of a truncated wave or disturbance. In relativity, this is always less than
3
or equal to c.
The wavenumber and the wavelength are related through k = 2π/λ, and the angular
frequency and the frequency are related through ω = 2πν. The so-called group velocity,
defined as the velocity at which the maximum of a wave packet moves through space, can
be found from
dω
,
(4)
dk
and as we shall see, this velocity is equal to the velocity of the particle, vgroup = v. Thus,
vgroup =
for De Broglie material waves, vphase vgroup = c2 .4
In the next section we apply a full covariant formalism, based on the Minkowski relativistic four-vector description of particles and waves, to show that ‘wave-particle duality’ in
relativity naturally leads to the De Broglie relation. The compliance of the De Broglie equation with the principle of relativity is analyzed in Sect. III, where we also explicitly show that
the phase and group velocities actually do transform as ordinary velocities under Lorentz
boosts. In fact, the way the phase and the group velocities actually transform is crucial to
ensure the De Broglie hypothesis covariance, and this is shown explicitly. In Sect. IV we
draw our conclusions, stressing the genuine relativistic character of the De Broglie relation.
II.
RELATIVISTIC DESCRIPTION FOR PARTICLES AND FOR WAVES
The wave-particle duality is an intrinsically relativistic concept, with no correspondence
in classical non-relativistic physics.2 In this section we explore the four-vector description of
a massive particle and of a plane wave, to conclude that the De Broglie relation naturally
emerges from such a formalism.
A.
Four-vector description of a particle
Let us consider a free particle with inertia m, moving with velocity ~v = (vx , vy , vz ) in
reference frame S. The linear momentum, p~ = (px , py , pz ), and the energy, E, are p~ = γm~v ,
and E = γmc2 , respectively, and for this particle, relations hold that
~v =
c2
p~,
E
(5)
and
E 2 = c2 p~ · p~ + m2 c4 .
4
(6)
The momentum-energy four-vector (in energy units), is (the time-like component is real and
it is the fourth component of the four-vector)



c px
c γ m vx



 cp 
 cγ mv
y
 y

µ
(E ) = 
 = (E) = 
 c pz 
 c γ m vz



E
γ m c2




.


(7)
The norm of this four-vector is a relativistic invariant,10 expressing the information already
provided by Eq. (6), we obtain Eµ E µ = E 2 − c2 p2 = m2 c4 (we are using a diagonal metric
tensor with g44 = −gii = 1). From Eq. (6) we may write E dE = c2 p~ · d~p , or using (5),
dE = ~v · d~p ,
(8)
which is equivalent to d (γc2 ) = ~v · d(γ~v ). From Eq. (8), the components of the velocity are
vi = ∂E/∂pi , exactly as in non-relativistic mechanics for a free particle.
B.
Four-vector description of a wave
A harmonic plane wave of amplitude A is described by
i
h ~
ψ(x, y, z, t) = A exp i k · ~r − ωt ,
(9)
where ~k is the wavenumber and ω the angular frequency, both already introduced in Sect. 1.
For this wave one usually considers the four-vector k µ (with wavenumber units),11 which is
explicitly written below, together with the contravariant position-time four-vector as




kx
x




 k 
 y 
 y 


(k µ ) = 
(10)
 and (xµ ) = 
.
 kz 
 z 




ω/c
ct
One recognizes that the exponent in Eq. (9) is the internal product in Minkowski space of
these four-vectors. Hence it is a relativistic invariant that12
xµ k µ = −xkx − yky − zkz + ωt = φ,
(11)
with φ being the phase of the wave, which is the same for all inertial observers. For a given
wavenumber ~k (hence, also a given ω) the internal product dxµ k µ vanishes or, ωdt−~k·d~r = 0.
5
This is the condition for constant phase , or φ = constant, meaning dφ = 0 (which is a frame
independent equation). From this condition, the phase velocity in frame S calculated from
~vphase = d~r/dt for the wave Eq. (9), can be related to the wavenumber and to the angular
frequency in that frame or
ω = ~vphase · ~k .
(12)
This relation generalizes the more familiar expression ω/k = λν = vphase .
C.
Relativistic De Broglie equation
According to Einstein’s wave-particle duality hypothesis for the light, one may consider
the four-vector (Ekµ ) = c~(k µ ) for the wave (in energy units) and the four-vector of Eq. (7),
(E µ ), for the particle with the linear momentum given by Eq. (2) (i.e. with the inertia given
by hν/c2 ). Then, the “wave–particle duality,” expressed by Ekµ = E µ , states that

 

c ~ kx
c (hν/c) `x

 

 c ~ k   c (hν/c) ` 
y
y

 


=
,
 c ~ ky   c (hν/c) `z 

 

~ω
hν
(13)
where ~` = (`x , `y , `z ) is the unit vector pointing in the photon’s direction.
The De Broglie hypothesis for a particle of mass m consists in expressing the “wave–
particle duality” again by Ekµ = E µ , which now implies13

 

c ~ kx
c γ m vx

 

 c~k   cγ mv 
y
y

 


=
.
 c ~ ky   c γ m vz 

 

2
~ω
γ mc
(14)
This four-vector equation can be regarded as the relativistic generalization14 of the simpler
De Broglie relation Eq. (1). Indeed, the four-vector Eq. (14) provides vector and scalar De
Broglie relations,15 namely
~~k = p~ = γ m ~v
(15)
~ω = E = γ m c2 .
(16)
and
6
Taking the modulus of (15), one may write this pair of relations as
λ=
h
h
=
p
γ mv
(17)
and
γ m c2
E
=
.
ν=
h
h
(18)
From Eq. (12) and from the De Broglie four-vector relation Eq. (14), the phase velocity
is such that
~vphase · ~v = c2 ,
(19)
an expression that generalizes Eq. (3).
A dispersion relation (i.e. a connection between a wave’s frequency and its wavenumber)
for matter waves16 follows from Eq. (6) and from the previous De Broglie relations Eq. (14).
In fact, the norm of (Ekµ ) allows us to write
m2 c4
ω 2 = c2~k · ~k +
.
~2
(20)
From this dispersion relation one obtains ω dω = c2 ~k · d~k and, after expressing ~k and ω as
functions of the velocity of the particle, as given by (14), results in
dω = ~v · d~k .
(21)
The group velocity is found from Eq. (4) and, more generally, satisfies dω = ~vgroup · d~k.
Therefore, one concludes that the group velocity associated with the matter wave is the
particle velocity ~vgroup = ~v , at least when the group velocity is aligned with ~k. Using the
energy and the linear momentum, one may also write down dE = ~vgroup · d~p and a direct
comparison with Eq. (8) allows us to conclude that the group velocity is the particle velocity
indeed. For an electromagnetic wave (or photon), the phase and the group velocities are the
same, as in vphase = vgroup = c.
A remark on the wave equation for a relativistic massive particle, corresponding to Eq. (9),
is pertinent here. From that equation and the De Broglie relations one has
i
ψ(x, y, z, t) = A exp − (Et − p~ · ~r) .
~
(22)
The exponent in the exponential is reference frame invariant since it corresponds to an internal product in Minkowski space, xµ pµ /~ = φ (where pµ = c−1 E µ ). The condition for
7
obtaining the phase velocity, or the condition for constant phase, is also frame invariant,
as per dxµ pµ = 0. Equation (22) is the plane wave solution of the Klein-Gordon equation
for a free particle.3 The Klein-Gordon equation is the relativistic quantum mechanical equation that applies to wave components of particles, and was established from the relativistic
connection between E and p~, whereas the Schrödinger equation was established from the
non-relativistic connection between energy and linear momentum.
III.
RELATIVISTIC TRANSFORMATIONS
One of the advantages of the Minkowski’s four-vector formalism is the equation covariance
under Lorentz transformations. This means given a four-vector equation in reference frame
S, such as Ekµ = E µ , in reference frame S 0 , the same equation holds, or Ek0µ = E 0µ . In
particular, the four-vector De Broglie relation of Eq. (14) is valid in any reference frame13
and, in S 0 , is explicitly given by

c ~ kx0

 c ~ k0

y

 c ~ k0
y

~ ω0


c γ 0 m vx0
 
  c γ 0 m v0
 
y
=
  c γ 0 m v0
z
 
γ 0 m c2




,


(23)
where the primed quantities refer to S0 , in particular γ 0 = γ(v 0 ).
In the asynchronous formulation of relativity17 the four-vectors in S and S0 are related
through the Lorentz transformation matrix. For the standard configuration (S0 moving with
respect to S with velocity V along the common x-x0 direction) this matrix is
−1/2
where γV = (1 − βV2 )

γV
0 0 −βV γV




L(V ) = 


0
1 0
0
0
0 1
0



.


−βV γV 0 0
γV
(24)
and βV = V /c. The contravariant four-vector transformation rule
is
µ
E 0 = Lµν (V ) E ν ,
(25)
and it applies to any other four-vector. For a particle moving in S with 3-momentum p~, the
8
momentum-energy transformation is
p0 x = γV px − V c−2 E ,
p0 y = py ,
p0 z = pz and
(26)
E 0 = γV (E − V px ) .
A similar set of relations holds for the wavenumber components and the angular frequency
(with the replacements p0i , → ki0 , pi →, ki and E 0 → ω 0 , E → ω).
For the sake of simplicity we shall restrict ourselves to the case of a particle moving along
the x (or the x0 ) direction, though the generalization to the more general case is straightforward (but tedious). Hence, for a particle with p~ = (px , 0, 0) the following expressions,
relating the wavelength and the frequency in S0 with the wavelength in S,
h
h
γV−1 λ
λ = 0 =
=
p
γV (px − c−2 E V )
1 − vphase,x V /c2
γV (E − px V )
E0
=
= γV λ−1 (vphase,x − V ) ,
ν0 =
h
h
0
(27)
(28)
then hold, where use has been made of the relations Eqs. (3), (5) and (26). Since the phase
velocity in S0 should be given by v 0 phase,x = λ0 ν 0 , one concludes that
v 0 phase,x =
vphase,x − V
,
1 − vphase,x V /c2
(29)
which is the ordinary x−component velocity transformation18 for the standard configuration.
This result is not surprising because the phase velocity in S0 is defined as ~v 0 phase = d~r 0 /dt0 ,
and both d~r 0 and dt0 transform in the standard way under Lorentz transformations as
given by Eq. (25), which is like Eq. (26), with the replacements p0i → dx0i , pi → dxi and
E 0 /c → cdt0 , E/c → cdt.
Should one consider a particle moving in any direction, the usual expressions for the y
and z components of the transformed phase velocity would also be obtained, namely
v 0 phase,i =
γV−1 vphase,i
for i = y, z .
1 − vphase,x V /c2
(30)
One should note that the phase velocity can be found from the frame independent equation
ω 0 dt0 − ~k 0 · d~r 0 = 0, giving ~v 0phase = d~r 0 /dt0 is the phase velocity in S0 . Therefore, the phase
velocity transforms like a normal velocity under Lorentz transformations, as also shown in
ref.19 .
9
The group velocity is equal to the particle velocity. Therefore the group velocity also
transforms in the same way, like an ordinary velocity, or
v 0 group,x =
vgroup,x − V
1 − vgroup,x V /c2
v 0 group,i =
γV−1 vgroup,i
for i = y, z .
1 − vgroup,x V /c2
(31)
Now, it can be directly checked that, provided Eq. (12) holds in S, ω 0 = ~v 0phase · ~k 0 must hold
as well in S0 ; and, provided Eq. (21) holds in S0 , with ~v = ~vgroup , dω 0 = ~v 0
· d~k 0 must also
group
hold in S0 . Moreover, from Eqs. (29)–(31) we may explicitly show that Eq. (19) is also valid
in S0 :
~v 0phase · ~v 0group = c2 ,
(32)
provided ~vphase · ~vgroup = c2 in S, as is actually the case for a De Broglie matter wave. The
dot product of two 3-vectors is not a Lorentz invariant, in general, but remarkably the dot
product of the phase and the group velocity for a De Broglie wave is a frame independent
result. In fact, a Lorentz transformation (along the x−axis) acting on two velocities ~u and
~v gives
~u 0 · ~v 0 = u0x vx0 + u0y vy0 + u0z vz0
=
(ux − V )(vx − V ) + (~u · ~v − ux vx ) [1 − (V /c)2 ]
.
(1 − ux V /c2 )(1 − vx V /c2 )
(33)
If and only if, in frame S, ~u · ~v = c2 , will the above become ~u 0 · ~v 0 = c2 , making this
dot product a Lorentz invariant. We stress that, in order to obtain this important result,
the phase and the group velocities should transform exactly in the same way, as ordinary
velocities. This ensures the covariance of Eqs. (12) and (21), and the frame invariance of
Eq. (19). In sum, only the proper velocity transformations of both the phase and the group
velocity lead to the frame independent result ~vphase · ~vgroup = ~v 0phase · ~v 0group = c2 .
We should note that, while both the phase and group velocities transform as ordinary
velocities for this special case, in general they do not transform in the same way for all wave
phenomena.18
IV.
CONCLUSIONS
The De Broglie relation is usually presented in university courses before the students
get acquainted with relativity and particularly with the four-vector formalism in Minkowski
space. But when students are familiarized with this more advanced relativity formalism,
10
it is a good opportunity to re-visit the De Broglie relations, as we propose in this article.
In our view, the relativistic formalism brings more physical insight to that basic quantummechanical relation. Note that we do not criticize the way the De Broglie relation is usually
introduced, rather we claim that it is worthwhile to revisit the matter after the students
are familiarized with advanced relativistic topics, including the covariance of four-vector
equations under Lorentz transformations. As we have shown in this paper, relativity puts,
in a better perspective, the scalar and the vector content of the four-vector equation that
expresses the De Broglie’s idea. On the other hand, relativity also allows for a more clear
understanding of the wave-particle duality both when we refer to light or to matter.
Moreover, in this paper we present in vector form certain relations that are usually
presented as scalar ones, such as the relation between linear momentum and the wavenumber.
This is almost automatically accomplished by working with four-vectors. On the other
hand, the four-vector formalism embodies covariance under Lorentz transformations, and
this facilitates finding the De Broglie relations in different inertial reference frames. In the
framework of the four-vector formalism, the way the De Broglie relation is expressed in
different frames is self-evident. This is not the case in the usual presentation of the De
Broglie relation in textbooks.
The relativistic context is also an opportunity to discuss the phase and the group velocity
(this is the particle’s velocity) as presented in this paper. We showed that ~vphase · ~vgroup = c2
is frame independent only if the phase and the group velocities transform under Lorentz
transformations as normal velocities. The expression can be generalized to any two velocities
whose dot product is c2 in a certain inertial frame.
We deem that there is an unquestionable pedagogical advantage in discussing the De
Broglie relation in a relativistic framework and this can be done, with great advantage
to the students, in an intermediate course on relativity (sometimes integrated in a second
Electromagnetism curricular unit) or even in an advanced quantum mechanics course, where
the interrelations between quantum mechanics and relativity are presented and explored.
According to own experience as instructors, by placing the De Broglie relation in a fourvector perspective, when the students are becoming familiar with the formalism of special
relativity, brings them a new physical insight both on the powerfulness of the relativistic
11
four-vector formalism and on the significance of the De Broglie relation.
∗
[email protected]
†
[email protected]
‡
[email protected]
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13